CK-12 Basic Probability and Statistics - A Short Course

http://www.ck12.org Chapter 5. The Shape, Center and Spread of a Normal Distribution - Basic

1. To determine the measure of each value from the mean, subtract the mean of the data from each value in the
   data set.(x−x ̄)


2. Square each of these differences and add the positive, squared results.


3. Divide this sum by the number of values in the data set.

These steps for calculating the variance of a data set can be summarized in the following formula:

σ^2 =∑

(x−x ̄)^2

n

where:

xrepresents the data value; ̄xrepresents the mean of the data set;nrepresents the number of data values. Remember
that the symbol∑stands for summation.

Example 1:

Given the following weights (in pounds) of children attending a day camp, calculate the variance of the weights.

52 , 57 , 66 , 61 , 69 , 58 , 81 , 69 , 74

TABLE5.1:

x (x−x ̄) (x−x ̄)^2

52 -13.2 174.24

57 -8.2 67.24

66 0.8 0.64

61 -4.2 17.64

69 3.8 14.44

58 -7.2 51.84

81 15.8 249.64

69 3.8 14.44

74 8.8 77.44

x ̄=

∑(x)

n

σ^2 =

∑(x−x ̄)^2

n

x ̄=

587

9

σ^2 =

667. 56

9

x ̄= 65. 2 σ^2 = 74. 17

Remember that the variance is the mean of the squares of the differences between the data value and the mean of the
data. The resulting value will take on the units of the data. This means that for the variance of the data above, the
units would be square pounds.

The standard deviation is simply the square root of the variance for the data set. When the standard deviation is
calculated for the above data, the resulting value will be in pounds. This table could be extended to include a
frequency column for values that are repeated adding three additional columns to the table. This often leads to errors
in calculations. Since simple is often best, values that are repeated can just be written in the table as many times as
they appear in the data.